id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-9930fca9e21a22e03696955513775b3a39d3a588c7f5ba781f7fd2dd31e2813c | At a dinner party, 20 geniuses are seated around a round table. Three specific guests, Brainy Brian James, Danger Dumie, and King Kwan, must not sit next to each other and must have at least three seats separating each from the other two. If the number of valid seating arrangements is $p(q!)$ where $p$ is prime and $|p... | [
"23"
] | [
"At a dinner party, 20 geniuses are seated around a round table. Three specific guests, Brainy Brian James, Danger Dumie, and King Kwan, must not sit next to each other and must have at least three seats separating each from the other two. If the number of valid seating arrangements is $p(q!)$ where $p$ is prime an... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-d40f2e8f3e3efd325cb5b76f6ab9926972556cce81d593f445fe72e3f6497198 | At a dinner party, Alex and Charlie invite four couples. Each person shakes hands only with the people they have not met before. After the party, Alex asks Charlie and the eight guests how many hands they shook, and receives nine different answers. How many people did Charlie shake hands with? | [
"4"
] | [
"At a dinner party, Alex and Charlie invite four couples. Each person shakes hands only with the people they have not met before. After the party, Alex asks Charlie and the eight guests how many hands they shook, and receives nine different answers. How many people did Charlie shake hands with?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b38fac0be45a4b98a757f6bcf7540884325aad4dab8dcb93a87e91c752b19b0 | At a dinner party, there are $N$ hosts and $N$ guests seated around a circular table, where $N \geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Calculate the minimum number ... | [
"8"
] | [
"At a dinner party, there are $N$ hosts and $N$ guests seated around a circular table, where $N \\geq 4$. A pair of two guests will chat with one another if either there is at most one person seated between them or if there are exactly two people between them, at least one of whom is a host. Calculate the minimum n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-ddce125c6f3d0a062d19ef3cc6115b01938be90f57dd6a3e2db3d49a55b2fc38 | At a dinner, 50 mathematicians each order 1 of 5 possible entrees, and there is at least one order of each entree. The cost of entree \(n\) is equal to the number of people who order entree \(n+1\), where entree 6 is entree 1. What is the minimum total cost of the dinner? | [
"95"
] | [
"At a dinner, 50 mathematicians each order 1 of 5 possible entrees, and there is at least one order of each entree. The cost of entree \\(n\\) is equal to the number of people who order entree \\(n+1\\), where entree 6 is entree 1. What is the minimum total cost of the dinner?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-168b1172d2eedb1ee97a6812d0b557e8e0de8ff4dd17230c16433ff15c04c025 | At a distance of 10m from a speaker, the intensity of the sound is \(10^{-2} \, \text{W/m}^2\). What will be the ratio of the sound levels in dB at the original distance and the distance where the intensity is reduced by a quarter? | [
"\\(-6.02\\) dB"
] | [
"At a distance of 10m from a speaker, the intensity of the sound is \\(10^{-2} \\, \\text{W/m}^2\\). What will be the ratio of the sound levels in dB at the original distance and the distance where the intensity is reduced by a quarter?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d742811acfe4ac7370a2466db7a2722729aa20856321143294ee9f016976bd8e | At a fish market with 10 stalls, each stall sells 10 kinds of fish, with each kind coming from either the North Sea or the Mediterranean Sea. A number $k$ of customers each buy one fish from each stall, ensuring they get exactly one of each kind of fish. Additionally, for each pair of customers, there is at least one k... | [
"1024"
] | [
"At a fish market with 10 stalls, each stall sells 10 kinds of fish, with each kind coming from either the North Sea or the Mediterranean Sea. A number $k$ of customers each buy one fish from each stall, ensuring they get exactly one of each kind of fish. Additionally, for each pair of customers, there is at least ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-e74e25bfbe2607af8d0a3da88eeb9dd458fe1b45dfbce9787c8d63a8103e19cc | At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw, and zero points for the defeat. At the end of the tournament, the sum of the winning points of all teams was 50. How many teams participated in this tournament? | [
"10"
] | [
"At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw, and zero points for the defeat. At the end of the tournament, the sum of the winning points of all teams was 50. How many teams participated in this tournament?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26a6c5246e6467c6681933d0dfeef55936ced39c88333b966177454230fedee4 | At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw, and zero points for the defeat. At the end of the tournament, the sum of the winning points of all teams was 50. What is the difference between the highest number of points won by any te... | [
"18"
] | [
"At a football tournament, each team plays with each of the remaining teams, winning three points for the win, one point for the draw, and zero points for the defeat. At the end of the tournament, the sum of the winning points of all teams was 50. What is the difference between the highest number of points won by a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-331a190637c86659e55b78a2b2d85a8f4f11a465df12ab08cbe63eb7d7829e1e | At a gala banquet, $12n + 6$ chairs, where $n \in \mathbb{N}$, are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposit... | [
"4n + 2"
] | [
"At a gala banquet, $12n + 6$ chairs, where $n \\in \\mathbb{N}$, are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0fb0a31b2ebe177caba40e6d70b3c6dd0fb8274904f683d9d82cf2465f32fa9 | At a gala banquet, $12n+6$ chairs are equally arranged around a large round table. A seating is called a proper seating of rank $n$ if $6n+3$ opposite-sex couples sit around the table such that each seated person also has exactly one sibling of the opposite sex present (siblings cannot be married to each other) and eac... | [
"4n + 2"
] | [
"At a gala banquet, $12n+6$ chairs are equally arranged around a large round table. A seating is called a proper seating of rank $n$ if $6n+3$ opposite-sex couples sit around the table such that each seated person also has exactly one sibling of the opposite sex present (siblings cannot be married to each other) an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-fcff5b7bc1411fb577208cd44f1b7b4a46faa151bcdcdc3d502ff4383b573b21 | At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? | [
"245"
] | [
"At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8f02c0c5ee158d3550f04af74e4ae081430ec68bdc65eed1ddc6f42eec907527 | At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur? | [
"245"
] | [
"At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 6
} | 38 | 0 | 0 |
omr-ws-2b98c8d0904db9ea74200a053e54bf4e1fffd2db7cae070815ba7ca9217c5622 | At a high school ball dinner, each boy knows at least 6 girls and each girl knows a maximum of 8 boys. If there are $b$ boys and $g$ girls, find the minimum value of the ratio $\frac{\text{number of boys able to dance with a known girl}}{b}$ given that $b = 2g$. | [
"\\dfrac{3}{4}"
] | [
"At a high school ball dinner, each boy knows at least 6 girls and each girl knows a maximum of 8 boys. If there are $b$ boys and $g$ girls, find the minimum value of the ratio $\\frac{\\text{number of boys able to dance with a known girl}}{b}$ given that $b = 2g$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-c8906bd831cd7c0ac027a91e08ceed0566d01d82dd8aab5dfb65771e1c92fee6 | At a hospital with 100 patients, each patient has at least one of the following ailments: a cold, the flu, or an earache. There are 38 patients with a cold, 40 with the flu, and an unknown number with earaches. Additionally, 17 patients have both a cold and an earache, and 7 patients have all three ailments. How many p... | [
"46"
] | [
"At a hospital with 100 patients, each patient has at least one of the following ailments: a cold, the flu, or an earache. There are 38 patients with a cold, 40 with the flu, and an unknown number with earaches. Additionally, 17 patients have both a cold and an earache, and 7 patients have all three ailments. How m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-90068b6e3ed06a8aca8e146e5875893ab6052cda85008c208cc28cba09d217ba | At a local school, 53.7802% (rounded) of the children are girls. What is the minimum number of children at the school? | [
"500000"
] | [
"At a local school, 53.7802% (rounded) of the children are girls. What is the minimum number of children at the school?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-a1715072910e0057544a78a1bfb9e19235442fb0254647ee8afc6f409a976350 | At a math camp, every $m$ students share exactly one common friend, where $m \ge 3$. Suppose person $P$ has the largest number of friends. Determine the number of friends person $P$ has. | [
"\\( m \\)"
] | [
"At a math camp, every $m$ students share exactly one common friend, where $m \\ge 3$. Suppose person $P$ has the largest number of friends. Determine the number of friends person $P$ has."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-6d6903bfaaded3ccf0ef9bab71f39e7a6da33788107927930cef3ad111b4dbb1 | At a math camp, every $m$ students share exactly one common friend, where $m \geq 3$. Suppose person $P$ has the largest number of friends. Determine the number of friends $P$ has. | [
"\\( m \\)"
] | [
"At a math camp, every $m$ students share exactly one common friend, where $m \\geq 3$. Suppose person $P$ has the largest number of friends. Determine the number of friends $P$ has."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-93e91537092be8217b69e1ecee07cffc1120a6f4507d1ce71b0bf34cb1bc6b93 | At a math conference, there are 100 people who have each met 9 others, another 100 who have each met 8 others, and so on, down to 100 people who haven't met anyone else before. If two participants are chosen at random, what is the probability that they have met before? | [
"\\(\\frac{1}{222}\\)"
] | [
"At a math conference, there are 100 people who have each met 9 others, another 100 who have each met 8 others, and so on, down to 100 people who haven't met anyone else before. If two participants are chosen at random, what is the probability that they have met before?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-41608da49e41f49704813b25218367732f8396f6147c83a205f412a3f7af80d9 | At a math conference, there are 100 people who have met none of the others before, another 100 who have met eight of the others, and so on down to the final 100 who haven't met anyone else before. If two participants are chosen at random, what is the probability that they have met before? | [
"\\dfrac{2}{555}"
] | [
"At a math conference, there are 100 people who have met none of the others before, another 100 who have met eight of the others, and so on down to the final 100 who haven't met anyone else before. If two participants are chosen at random, what is the probability that they have met before?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-f15a6cf595a7b61d3ed3ac419eb74d0c9dd5683fad3c53eb3c28b4ec8fa9a5d8 | At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | [
"32"
] | [
"At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-4dc1c217b24c317b45854dc96e847b9fa26997f8dd713e1250a485c84ad0edd6 | At a math contest, 57 students are wearing blue shirts, and 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. How many pairs are there where both students are wearing yellow shirts? | [
"32"
] | [
"At a math contest, 57 students are wearing blue shirts, and 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. How many pairs are there where both students are wearing yellow shirts?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-2c06ea93a833d5eb66a36963177c02d191f122deb8c41628ffaf4301a8a17fe0 | At a math contest, 57 students are wearing blue shirts, and 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | [
"32"
] | [
"At a math contest, 57 students are wearing blue shirts, and 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-fdf32cba74664121da04007ce79f439027b05bf0e6822c4b18a7e97775d62f94 | At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts? | [
"32"
] | [
"At a math contest, 57 students are wearing blue shirts, and another 75 students are wearing yellow shirts. The 132 students are assigned into 66 pairs. In exactly 23 of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e14059b20e3d0f60d7961cee96e8a4b4453ec02dd9f10d40be58d94694820de2 | At a math contest, each participant met at least 3 other participants who they already knew. Suppose the Jury selects an even number of participants (more than two) and arranges them around a table so that each participant is seated between those they know. Let $N$ be the minimum possible number of participants in such... | [
"4"
] | [
"At a math contest, each participant met at least 3 other participants who they already knew. Suppose the Jury selects an even number of participants (more than two) and arranges them around a table so that each participant is seated between those they know. Let $N$ be the minimum possible number of participants in... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-8e82a33e5e0c603914bfbac1cdad1d135ff63aee63994df60659bd762ab4a631 | At a math contest, each participant met at least 3 pals who they already knew. The Jury decides to choose an even number of participants (more than two) and arrange them around a table so that each participant is seated between people they know. Let $N$ be the minimum number of participants the Jury can choose to satis... | [
"6"
] | [
"At a math contest, each participant met at least 3 pals who they already knew. The Jury decides to choose an even number of participants (more than two) and arrange them around a table so that each participant is seated between people they know. Let $N$ be the minimum number of participants the Jury can choose to ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-9389ce90b8e075b43e444f744be5708959b8a94d57d33f01b280f8fe44c3eaec | At a math contest, there are $2n$ students, each submitting a problem. The contest is considered fair if there are $n$ participants who receive their problems from the other $n$ participants. Let $f(n)$ be the number of distributions of the problems to achieve a fair contest. Calculate the value of $f(5)$. | [
"30240"
] | [
"At a math contest, there are $2n$ students, each submitting a problem. The contest is considered fair if there are $n$ participants who receive their problems from the other $n$ participants. Let $f(n)$ be the number of distributions of the problems to achieve a fair contest. Calculate the value of $f(5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8f36beadca3b8604a6941307b27c60247a133004f3354808320649f52cf06fb7 | At a math olympiad, 72 medals were handed out. The total price of the medals was a 5-digit number with the three middle digits all being 9. If each medal cost the same integer amount, what was the price of each medal? | [
"1111"
] | [
"At a math olympiad, 72 medals were handed out. The total price of the medals was a 5-digit number with the three middle digits all being 9. If each medal cost the same integer amount, what was the price of each medal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-9d0ac35a5746e0d1c7bdc2d5527960ea2feb67f0ef20f0c367dcdbcd9838f756 | At a mathematical competition, $n$ students work on 6 problems, each with three possible answers. For every pair of students, the number of problems for which they have the same answers is either 0 or 2. Find the maximum possible value of $n$. | [
"9"
] | [
"At a mathematical competition, $n$ students work on 6 problems, each with three possible answers. For every pair of students, the number of problems for which they have the same answers is either 0 or 2. Find the maximum possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-ba4527483dae9e86affb3ba794ebd2c4a0160e5fa23159a9c9b446cbd7a8b010 | At a mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. If there are 10 mathematicians at the conference, calcul... | [
""
] | [
"At a mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. If there are 10 mathematicians at the conference, c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-4e19c5346aca14dc115a0eacfaa4505087027dd12fada61a8064151a783516f0 | At a mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. If there are 10 mathematicians at the conference, calcul... | [
"512"
] | [
"At a mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. If there are 10 mathematicians at the conference, c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e15686272c5579f5192c5b7992dd602415d698ca2095141ab9a7bd53095b4db6 | At a mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. If there are 10 mathematicians at the conference, find t... | [
"55"
] | [
"At a mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. If there are 10 mathematicians at the conference, f... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e03291723bcbd5fc24d77f06bf4e487c8c58de273abfc11a1a7fde223a76a88f | At a mathematical congress, it is observed that if one participant is placed in a single room, the remaining participants can be paired into double rooms where each pair knows each other. Let $P$ be a participant at the congress. Define $f(P)$ as the minimum number of participants, excluding $P$, that can form a round ... | [
"1"
] | [
"At a mathematical congress, it is observed that if one participant is placed in a single room, the remaining participants can be paired into double rooms where each pair knows each other. Let $P$ be a participant at the congress. Define $f(P)$ as the minimum number of participants, excluding $P$, that can form a r... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-bda994450c82e33e089a1afcf3c4d94117b2fe6b1532eb29e6bfc853981dcd7f | At a mathematical olympiad, eight problems were given to 30 contestants. The points for each problem are assigned based on the number of contestants who did not solve it: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. Ivan got fewer points than any other contestant. Determine the greates... | [
"58"
] | [
"At a mathematical olympiad, eight problems were given to 30 contestants. The points for each problem are assigned based on the number of contestants who did not solve it: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. Ivan got fewer points than any other contestant. Determine the gr... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ccb198246875cf3319ec5110e441b976f0d6766765d20de3b4601bfee16e7365 | At a meeting attended by 96 Thai delegates and an unknown number of Yaranikian delegates, it was discovered that for any selection of at least 69 Thai delegates and all Yaranikian delegates known to the selected Thais, there is at least one Yaranikian who knows more Thais than any other Yaranikian in the selection. Let... | [
"28"
] | [
"At a meeting attended by 96 Thai delegates and an unknown number of Yaranikian delegates, it was discovered that for any selection of at least 69 Thai delegates and all Yaranikian delegates known to the selected Thais, there is at least one Yaranikian who knows more Thais than any other Yaranikian in the selection... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05b6d3904a30d90fb741695d1584005b28999844fe1f6043e3506f13cade0c12 | At a meeting consisting of 2007 people, everyone shook hands with at least one other person. After the meeting, people who shook hands with no more than 10 people were given a red shirt. People who shook hands only with those who were given red shirts were given blue shirts. Calculate the maximum possible ratio of the ... | [
"\\dfrac{608}{61}"
] | [
"At a meeting consisting of 2007 people, everyone shook hands with at least one other person. After the meeting, people who shook hands with no more than 10 people were given a red shirt. People who shook hands only with those who were given red shirts were given blue shirts. Calculate the maximum possible ratio of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-3e8527a355c391a9799a990b20cf207bfb09c87607a505569abacc18d92c9a45 | At a meeting of $12k$ people, each person exchanges greetings with exactly $3k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting? | [
"36"
] | [
"At a meeting of $12k$ people, each person exchanges greetings with exactly $3k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-ebd24c14749550b480ff08c0baafde1994cb5dc592c41eb35a790230c062e943 | At a meeting of $18k$ people, each person exchanges greetings with exactly $3k + 6$ others. For any three people, there exist two of them such that the number of persons who exchanged greetings with both is the same (denote $n$). Find $k$. | [
"1"
] | [
"At a meeting of $18k$ people, each person exchanges greetings with exactly $3k + 6$ others. For any three people, there exist two of them such that the number of persons who exchanged greetings with both is the same (denote $n$). Find $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-b39a19b3c5f32ee4361ff726eee6a2a95a5ab4d4d98334646226a50e8c8a3e52 | At a meeting of $2021$ people, one person has no friends, another person has exactly one friend, and among any $4$ people, at least $2$ are friends. Let $N$ be the number of people who are all friends with each other. Calculate the value of $N$. | [
"2019"
] | [
"At a meeting of $2021$ people, one person has no friends, another person has exactly one friend, and among any $4$ people, at least $2$ are friends. Let $N$ be the number of people who are all friends with each other. Calculate the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-88111fbe68ea255f2300abc5d5e051fe77ac9cb86811570013c0fe39c64433f5 | At a meeting of 12k people, each person exchanges greetings with exactly 3k+6 other people. For any two people, the number of people who exchanged greetings with both of them is the same. How many people are there in the meeting? | [
"36"
] | [
"At a meeting of 12k people, each person exchanges greetings with exactly 3k+6 other people. For any two people, the number of people who exchanged greetings with both of them is the same. How many people are there in the meeting?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-594a2deb18e2bc8cfe31763160d2111164c1c48a5dafc98cc71c3683a329365b | At a meeting of 5 couples, including the Zweinstein-Curie couple, each person shook hands with some of the others, excluding their own spouse and themselves. Everyone answered differently when asked how many hands they had shaken. How many hands did Marta shake? | [
"4"
] | [
"At a meeting of 5 couples, including the Zweinstein-Curie couple, each person shook hands with some of the others, excluding their own spouse and themselves. Everyone answered differently when asked how many hands they had shaken. How many hands did Marta shake?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20bb36dfe698f0707ea1c869f59e1d59932654bffcefc57eba2f546910a1bd7a | At a meeting of \( 12k \) people, each person exchanges greetings with exactly \( 3k + 6 \) others. For any two people, the number who exchange greetings with both is the same. Find the sum of all possible values of \( k \). | [
"3"
] | [
"At a meeting of \\( 12k \\) people, each person exchanges greetings with exactly \\( 3k + 6 \\) others. For any two people, the number who exchange greetings with both is the same. Find the sum of all possible values of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac12337f778ae68778590690a699a51d707e066c968fb6f3673e8fe4c2b93688 | At a meeting with $25$ businessmen, twelve businessmen drank coffee, and ten businessmen drank tea. Five businessmen drank both coffee and tea. How many businessmen drank neither coffee nor tea? | [
"8"
] | [
"At a meeting with $25$ businessmen, twelve businessmen drank coffee, and ten businessmen drank tea. Five businessmen drank both coffee and tea. How many businessmen drank neither coffee nor tea?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7fcbe140864ad4513321774cba0fadcf2fff5d970d7f138747e8750e99247dd8 | At a meeting with $n$ people and $n+1$ groups, each consisting of 3 members, where no two groups have all three members in common, determine the minimum number of pairs of groups that share exactly one member. | [
"3n - 6"
] | [
"At a meeting with $n$ people and $n+1$ groups, each consisting of 3 members, where no two groups have all three members in common, determine the minimum number of pairs of groups that share exactly one member."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6dac088355c97d83d4af3edbbee456f25cc866c5e47547088d94f980c6e095c4 | At a meeting with 5000 attendants, each of whom can speak at least one language, the administration wants to divide them into two groups: - Group A: There exists one language that can be used by all people in this group. - Group B: Any two people in this group do not use a common language. Determine the minimum number ... | [
"2"
] | [
"At a meeting with 5000 attendants, each of whom can speak at least one language, the administration wants to divide them into two groups:\n- Group A: There exists one language that can be used by all people in this group.\n- Group B: Any two people in this group do not use a common language.\nDetermine the minimum... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-36f54f4590acfb4579d57d94c1f9e1ac0a3f27e989eb0b6e8cae1c19145d2dca | At a meeting with \( n = 102 \) people, for each subset of 6 people, you will find two people who know each other. Determine the minimum number of people in a subset such that all people in this subset know each other. | [
"21"
] | [
"At a meeting with \\( n = 102 \\) people, for each subset of 6 people, you will find two people who know each other. Determine the minimum number of people in a subset such that all people in this subset know each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-5227441c2d3d96ef6278d4ffce76369d2620e4d46fa4ecc6aa531d967f82df09 | At a meeting with \( n = 252 \) people, for each subset of 6 people, you will find two people who know each other. Determine the minimum number of people in a subset such that all people in this subset know each other. | [
"6"
] | [
"At a meeting with \\( n = 252 \\) people, for each subset of 6 people, you will find two people who know each other. Determine the minimum number of people in a subset such that all people in this subset know each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-943430c19fc60215d5a5380e699864149c3f35e3931b6873226735d3a268a6e9 | At a meeting with \( n = 512 \) people, for each subset of 6 people, you will find two people who know each other. Determine the minimum number of people in a subset such that all people in this subset know each other. | [
"7"
] | [
"At a meeting with \\( n = 512 \\) people, for each subset of 6 people, you will find two people who know each other. Determine the minimum number of people in a subset such that all people in this subset know each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-f5539da99a0ca62ee9873023c75532f358ff1fbb7b50797d418f06a405b84846 | At a meeting, each person is familiar with 22 others. If two persons $A$ and $B$ know each other, they do not have any common friends among the remaining people. If two persons $A$ and $B$ do not know each other, there are exactly six common acquaintances among the remaining people. How many people were at the meeting? | [
"100"
] | [
"At a meeting, each person is familiar with 22 others. If two persons $A$ and $B$ know each other, they do not have any common friends among the remaining people. If two persons $A$ and $B$ do not know each other, there are exactly six common acquaintances among the remaining people. How many people were at the mee... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-8c1b2820f4889edba876fd631aca920a714dc7fdc545edd461a353fdb665630a | At a meeting, each person knows 22 others. If two people know each other, they do not have any common friends among the remaining people. For each pair of people who do not know each other, there are six common acquaintances among the remaining people. How many people were at the meeting? | [
"100"
] | [
"At a meeting, each person knows 22 others. If two people know each other, they do not have any common friends among the remaining people. For each pair of people who do not know each other, there are six common acquaintances among the remaining people. How many people were at the meeting?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b5c5b763a4bb4533bbb68a29934b55fa1e9cbc7deaf11b50bfb88af8863a4d6 | At a meeting, each person shakes hands with some number of other people. Let $n$ be the total number of people at the meeting. If $k$ represents the number of people who shook hands an odd number of times, find the remainder when $k$ is divided by $2$ for any possible meeting scenario. | [
"0"
] | [
"At a meeting, each person shakes hands with some number of other people. Let $n$ be the total number of people at the meeting. If $k$ represents the number of people who shook hands an odd number of times, find the remainder when $k$ is divided by $2$ for any possible meeting scenario."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e60bfe2144fd3cd56cf99b652dadd11d7f1fdff03837b8b31924edaf0fa6f09b | At a movie theater, $2n$ people stand in line to buy tickets. The ticket price is $5. Suppose $n$ of those people only have $5 bills and the remaining $n$ people only have $10 bills. In the beginning, no change is available at the ticket counter. In how many ways can the $2n$ people be arranged in the line so that nobo... | [
"\\(\\frac{\\binom{2n}{n}}{n+1}\\)"
] | [
"At a movie theater, $2n$ people stand in line to buy tickets. The ticket price is $5. Suppose $n$ of those people only have $5 bills and the remaining $n$ people only have $10 bills. In the beginning, no change is available at the ticket counter. In how many ways can the $2n$ people be arranged in the line so that... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-e2f5072544e2575e17e5c39c3f7ba508dc68f5c9f83a8a67f67f6d4aba2a5553 | At a movie theater, there are currently \( m+k \) people queuing to buy tickets, including \( m \) people holding 10 US dollars and \( k \) people holding 5 US dollars. Each movie ticket costs 5 dollars and the ticket agent does not have any money in hand at the beginning. How many ways can the ticket buyers line up so... | [
"\\(\\frac{(m+k)!}{m!k!} \\times \\frac{m-k+1}{m+1}\\)"
] | [
"At a movie theater, there are currently \\( m+k \\) people queuing to buy tickets, including \\( m \\) people holding 10 US dollars and \\( k \\) people holding 5 US dollars. Each movie ticket costs 5 dollars and the ticket agent does not have any money in hand at the beginning. How many ways can the ticket buyers... | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 10,
"tir": 28
} | 54 | 0 | 0 |
omr-ws-d6d7f55dbd47e60e30ded9c34562a9355fe9a434aa7a433f2a06ae3c23943e5c | At a movie theatre, $m+n$ people stand in line to buy tickets. The ticket price is 5 dollars. Suppose $m \ge n$, and $m$ people have 5 dollar bills and the remaining $n$ people have 10 dollar bills. At the beginning, no change is available at the ticket counter. In how many ways can the order of these $m+n$ people be a... | [
"\\dfrac{(m - n + 1)}{(m + 1)} \\dbinom{m + n}{n}"
] | [
"At a movie theatre, $m+n$ people stand in line to buy tickets. The ticket price is 5 dollars. Suppose $m \\ge n$, and $m$ people have 5 dollar bills and the remaining $n$ people have 10 dollar bills. At the beginning, no change is available at the ticket counter. In how many ways can the order of these $m+n$ peopl... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-d63b3dcc24fed00a4d887a130a509c3011ab682cbcbad9845a1bfb99541de004 | At a networking session, 13 pairs of twins and my twin brother and I are present. Each person exchanges business cards with a different number of people, excluding their twin sibling. How many people did my twin brother exchange business cards with? | [
"13"
] | [
"At a networking session, 13 pairs of twins and my twin brother and I are present. Each person exchanges business cards with a different number of people, excluding their twin sibling. How many people did my twin brother exchange business cards with?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11febacb94eff228b14b19265487764c10b22d4b8190469a4bf57af19bdd8b54 | At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 5 students are taking both classes. How many people are in the chemistry class? | [
"36"
] | [
"At a particular school with 43 students, each student takes chemistry, biology, or both. The chemistry class is three times as large as the biology class, and 5 students are taking both classes. How many people are in the chemistry class?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f7b78d0e77574dc4274eaf89110732c41bce15bc8d913c830eae5269fb734e2 | At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly o... | [
"7"
] | [
"At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \\cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exa... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-b2462c45d3620fbfa89a401f6126f4b79f43c1d1e338bd80ed7bd4ac33eab38c | At a party of 13 married couples, each man shook hands with everyone except his spouse and no handshakes took place between women. How many handshakes occurred among these 26 people? | [
"234"
] | [
"At a party of 13 married couples, each man shook hands with everyone except his spouse and no handshakes took place between women. How many handshakes occurred among these 26 people?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ccac39a214ef2ceea09307952ec726584b90724450b3dbcb4f4633db00cabcca | At a party of 20 people, 11 people are friends with everyone else, and the other 9 people each have exactly 13 friends at the party. Each person shakes hands with each of their friends. What is the total number of handshakes? | [
"190"
] | [
"At a party of 20 people, 11 people are friends with everyone else, and the other 9 people each have exactly 13 friends at the party. Each person shakes hands with each of their friends. What is the total number of handshakes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68d10b1c2125efbe1bc229e60a6b7a6bc5cd8756401af84e773545d79369b69f | At a party organized by a host and hostess, k couples are invited. Each person can shake hands with every other person except their spouse. The host asks everyone, including the hostess, how many handshakes they had and receives 2k+1 different answers. How many handshakes did the hostess have? | [
"1"
] | [
"At a party organized by a host and hostess, k couples are invited. Each person can shake hands with every other person except their spouse. The host asks everyone, including the hostess, how many handshakes they had and receives 2k+1 different answers. How many handshakes did the hostess have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e12e2b9175d84f8a24b88a98a69ab7dafefddd2440ccb393d9400bb4522c982 | At a party there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of the women. For which values of $r$ and $s$ must there be at least one pair of a man and a woman who like each other? | [
"\\( r + s > n \\)"
] | [
"At a party there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of the women. For which values of $r$ and $s$ must there be at least one pair of a man and a woman who like each other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 3
} | 51 | 0 | 0 |
omr-ws-818d8fdc8dad71b73bbae8598589b1480e0d16652932a80f92537494451ac447 | At a party with $12k$ people, each person shakes hands with exactly $3k+6$ others. For any two persons, the number of people who have shaken hands with both is the same. Find the number of people at the party. | [
"24"
] | [
"At a party with $12k$ people, each person shakes hands with exactly $3k+6$ others. For any two persons, the number of people who have shaken hands with both is the same. Find the number of people at the party."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-960b232f4d5214d51685126492f929ed90bd09f44c327e683ba762dbffa278fc | At a party with $2N$ people where each person knows at least $N$ others, determine the minimum number of distinct groups of four people that can be seated around a round table such that each person knows both of their neighbors. Express your answer in terms of $N$. | [
"\\dfrac{N(N-1)}{2}"
] | [
"At a party with $2N$ people where each person knows at least $N$ others, determine the minimum number of distinct groups of four people that can be seated around a round table such that each person knows both of their neighbors. Express your answer in terms of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-c22c50b883a0b32b78051e9ef4cb5d560f75370de4927b326b01453d3c79c731 | At a party with $2n$ individuals, where $n \geq 2$, each individual knows at least $n$ other individuals. Find the minimum number of groups of four individuals who can sit at a round table such that each individual knows their neighbors at the table, and let this number be $k$. If $n=5$, what is the value of $k$? | [
"3"
] | [
"At a party with $2n$ individuals, where $n \\geq 2$, each individual knows at least $n$ other individuals. Find the minimum number of groups of four individuals who can sit at a round table such that each individual knows their neighbors at the table, and let this number be $k$. If $n=5$, what is the value of $k$?... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-5028930b5265a3c3f8faa03b84d23e9c0c28047e3ffefd2d1123358d8943425e | At a party with $2n$ people where each person has an even number of friends, let $f(x, y)$ denote the number of common friends between person $x$ and person $y$. Find the sum of all possible values of $f(x, y)$ for a pair $(x, y)$ where $f(x, y)$ is even. | [
"n(n-1)"
] | [
"At a party with $2n$ people where each person has an even number of friends, let $f(x, y)$ denote the number of common friends between person $x$ and person $y$. Find the sum of all possible values of $f(x, y)$ for a pair $(x, y)$ where $f(x, y)$ is even."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-199fb1a6f00c1b7b7d535b522fe92b9034577aae0ba863c58bc7d3d10f13bd67 | At a party with $n \geq 2$ people where each person has at least one friend also at the party, let $f(n)$ be the minimum possible size of a group of people such that any person not in the group has a friend within the group. Find $f(100)$. | [
"50"
] | [
"At a party with $n \\geq 2$ people where each person has at least one friend also at the party, let $f(n)$ be the minimum possible size of a group of people such that any person not in the group has a friend within the group. Find $f(100)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-7bcde81d9026a01709e8990c70caee496543c1f14428d14892e7bdf9e79e3a4b | At a party with $n$ people, each person counts the number of other people they know. Let $S$ be the sum of all these counts. If $n = 2023$, find the remainder when $S$ is divided by $2$. | [
"0"
] | [
"At a party with $n$ people, each person counts the number of other people they know. Let $S$ be the sum of all these counts. If $n = 2023$, find the remainder when $S$ is divided by $2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-546059d91fc6203759807c8cc4bedd56bc96559fea02f020766e42411eccb53e | At a party with $n$ people, it is known that among any $4$ people, there are either $3$ people who all know one another or $3$ people none of which knows another. Suppose you are organizing the party and want to separate the $n$ people into two rooms such that everyone in one room knows one another and no two people in... | [
"4"
] | [
"At a party with $n$ people, it is known that among any $4$ people, there are either $3$ people who all know one another or $3$ people none of which knows another. Suppose you are organizing the party and want to separate the $n$ people into two rooms such that everyone in one room knows one another and no two peop... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-cc7c98dc1f6bed3dc931816bad62bc678b6cbee8fc2a93b9bbf65e37d3eaf1cc | At a party with $n$ people, let $f(n)$ be the minimum number of pairs of people who have exactly the same number of friends at the party. Calculate $f(10)$. | [
"2"
] | [
"At a party with $n$ people, let $f(n)$ be the minimum number of pairs of people who have exactly the same number of friends at the party. Calculate $f(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-a103e82aa770c546299f361c7f247f4521e71ecfec1feef4979ad21fef17e5dd | At a party with $v$ boys and $v$ girls, each boy danced with $k$ girls, and any two boys had exactly $d$ common (female) dance partners, where $d < k$. Given that $v = 10$ and $d = 2$, find the value of $k$ if it is known that each girl danced with exactly $k$ boys and any two girls had exactly $d$ common (male) dance ... | [
"6"
] | [
"At a party with $v$ boys and $v$ girls, each boy danced with $k$ girls, and any two boys had exactly $d$ common (female) dance partners, where $d < k$. Given that $v = 10$ and $d = 2$, find the value of $k$ if it is known that each girl danced with exactly $k$ boys and any two girls had exactly $d$ common (male) d... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c9e199e7250ed15b798daf1ff67fdb432bf78b21cf0fd7d3c9258722794c7150 | At a party with 10 couples (20 people), everyone except the host shakes hands with a different number of people, and no one shakes hands with their partner. How many people did the host's wife shake hands with? | [
"10"
] | [
"At a party with 10 couples (20 people), everyone except the host shakes hands with a different number of people, and no one shakes hands with their partner. How many people did the host's wife shake hands with?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbcc75a367307368589457897717a699f2842e5a23d5c80184c222bb804c2d27 | At a party with 100 people, in any set of 4 people, there is at least one person who is mutually acquainted with the other three. Given that there are three people who are mutually unacquainted with each other, calculate the number of possible sets of 3 people who are mutually unacquainted with each other in the remain... | [
"0"
] | [
"At a party with 100 people, in any set of 4 people, there is at least one person who is mutually acquainted with the other three. Given that there are three people who are mutually unacquainted with each other, calculate the number of possible sets of 3 people who are mutually unacquainted with each other in the r... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f9a8cb958703ecfabaa14ea7ddae1b8ff0c13237037ff9d6295fb9e31b48d7bb | At a party with 1000 people, the number of people who shook hands with at most 962 others is at most 37. Determine the minimum number of people, $n$, such that one can always find a group of $n$ people, all of whom have shaken hands with each other. | [
"27"
] | [
"At a party with 1000 people, the number of people who shook hands with at most 962 others is at most 37. Determine the minimum number of people, $n$, such that one can always find a group of $n$ people, all of whom have shaken hands with each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-c47687d608014b360406697238ca6be16cd4f04227a0ea0be9f7bbd4bb205c8f | At a party with 13 married couples (26 people), each man shakes hands with everyone except his spouse, and no handshakes take place between women. How many handshakes occurred? | [
"234"
] | [
"At a party with 13 married couples (26 people), each man shakes hands with everyone except his spouse, and no handshakes take place between women. How many handshakes occurred?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9f3646610c77cf21633d688ba8ce6a99eccb8e9141ffbe7d46290521d3e8be34 | At a party with 13 married couples (26 people), each man shakes hands with everyone except his spouse, and no handshakes take place between women. How many handshakes were there among these 26 people? | [
"234"
] | [
"At a party with 13 married couples (26 people), each man shakes hands with everyone except his spouse, and no handshakes take place between women. How many handshakes were there among these 26 people?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-651e7332b2bdeaf8d0a56e327cdf25bc95800a47383573ad003268829eec8492 | At a party with 2 married couples present, various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening, Mr. Jones asked everyone else, including his wife, how many hands he or she had s... | [
"1"
] | [
"At a party with 2 married couples present, various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening, Mr. Jones asked everyone else, including his wife, how many hands he or she ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36174a7a2a866afa21e46696be69bf24da49c6b8b5cf9d9202b68cb93e119c74 | At a party with 201 people from 5 different nationalities, each person's age is an integer between 1 and 100 inclusive. Given that in any group of six people, at least two have the same age, find the minimum number of people who must share the same nationality and age. | [
"9"
] | [
"At a party with 201 people from 5 different nationalities, each person's age is an integer between 1 and 100 inclusive. Given that in any group of six people, at least two have the same age, find the minimum number of people who must share the same nationality and age."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64a5714d9fecee44ee2671a41a1a0395f7b0d6991e37886ac2adae06647928e7 | At a party with 2010 boys and 2010 girls, each boy brings a bouquet of flowers and each girl brings a bar of chocolate. They sit in a circle and give their presents to the person on their right. Determine the number of ways to arrange the boys such that after several rounds, every boy ends up with a bar of chocolate an... | [
"\\(2^{2010}\\)"
] | [
"At a party with 2010 boys and 2010 girls, each boy brings a bouquet of flowers and each girl brings a bar of chocolate. They sit in a circle and give their presents to the person on their right. Determine the number of ways to arrange the boys such that after several rounds, every boy ends up with a bar of chocola... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-efbc5528a0cea6b05a72b89c37136b0f939d7760dfeea2572dc1803b85466b5c | At a party with 2015 couples, each husband knows their own wife and vice versa. Each person knows a different number of other people. How many people does the wife of a particular man know? (The "knowing of others" is symmetric but not reflexive.) | [
"2015"
] | [
"At a party with 2015 couples, each husband knows their own wife and vice versa. Each person knows a different number of other people. How many people does the wife of a particular man know? (The \"knowing of others\" is symmetric but not reflexive.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-caef173e18a99591cd13d4ef853dd36cf8ae811bdf84cbc23774a33df7729f15 | At a party with 25 IMO participants, each pair communicates in exactly one language, and for any group of three participants, at least one of them communicates with the other two in the same language. Let $N$ be the maximum number of participants that any single participant communicates with in the same language. Find ... | [
"12"
] | [
"At a party with 25 IMO participants, each pair communicates in exactly one language, and for any group of three participants, at least one of them communicates with the other two in the same language. Let $N$ be the maximum number of participants that any single participant communicates with in the same language. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-d24caa7c82da1c8b41c2dc6baf8ac400eb048a33a81fe6c2776e67b105f90c03 | At a party with 3 married couples present, various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening, Mr. Jones asked everyone else, including his wife, how many hands he or she had s... | [
"2"
] | [
"At a party with 3 married couples present, various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening, Mr. Jones asked everyone else, including his wife, how many hands he or she ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8449f6b76643b731a47f9b464e0f3b6278c94bc948ff30768945d28e8b8fbde3 | At a party with 4 other married couples (total of 10 people), everyone except one person shakes hands with some guests but not with their spouse. Jim and Jeri do not shake hands with each other. After the party, Jim asks each guest how many people they shook hands with, and he gets the answers 0, 1, 2, 3, 4, 5, 6, 7, 8... | [
"4."
] | [
"At a party with 4 other married couples (total of 10 people), everyone except one person shakes hands with some guests but not with their spouse. Jim and Jeri do not shake hands with each other. After the party, Jim asks each guest how many people they shook hands with, and he gets the answers 0, 1, 2, 3, 4, 5, 6,... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0c8011cb818a2b96b217aed1addc4c61ba810e0fbc289eabae86833746bb2b38 | At a party with 50 boys and 99 girls, each boy knows a certain number of girls. Suppose each boy invites exactly one girl for a dance. Determine the maximum number of boys who can be dancing with a girl they know, assuming that if a boy is not dancing with a known girl, then all the girls he knows are not dancing. | [
"50"
] | [
"At a party with 50 boys and 99 girls, each boy knows a certain number of girls. Suppose each boy invites exactly one girl for a dance. Determine the maximum number of boys who can be dancing with a girl they know, assuming that if a boy is not dancing with a known girl, then all the girls he knows are not dancing.... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-ed5666f159215123d9b353f2c67d602760f73bc6ba2534e2c8d38c0d429b19ec | At a party with 9 people, between any three of them there are two who know each other. Find the maximum number of groups of four people who all know each other. | [
"126"
] | [
"At a party with 9 people, between any three of them there are two who know each other. Find the maximum number of groups of four people who all know each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-84ed0c24cbdc645a7bd4139430141d746e3f37e36000b84d944adc76e0c8e530 | At a party with Alice, Bob, and 10 other couples (21 people total), nobody shook their partner's hand, and each of the 21 people shook a different number of hands. How many hands did Bob shake? | [
"10"
] | [
"At a party with Alice, Bob, and 10 other couples (21 people total), nobody shook their partner's hand, and each of the 21 people shook a different number of hands. How many hands did Bob shake?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55d83d3374df362dfeaf79b65ba8fec7d31a2820aa4c33a86b7cf8c4247ab1cd | At a party with Henry, his wife, and 4 other married couples, everyone shook hands with others except for their spouse and one person. The number of handshakes reported by the guests were 0, 1, 2, 3, 4, 5, 6, 7, and 8. How many people did Henry shake hands with? | [
"4"
] | [
"At a party with Henry, his wife, and 4 other married couples, everyone shook hands with others except for their spouse and one person. The number of handshakes reported by the guests were 0, 1, 2, 3, 4, 5, 6, 7, and 8. How many people did Henry shake hands with?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-694d3a34aa56d678555b2c4e7cfc83dceab246390a10492c24b64c8d7734bad3 | At a party with Jim, Jeri, and 4 other married couples, each person except one shakes hands with some guests but not with their spouse. Jim asks each guest how many people they shook hands with and gets answers 0, 1, 2, 3, 4, 5, 6, 7, 8. How many people did Jeri shake hands with? | [
"4"
] | [
"At a party with Jim, Jeri, and 4 other married couples, each person except one shakes hands with some guests but not with their spouse. Jim asks each guest how many people they shook hands with and gets answers 0, 1, 2, 3, 4, 5, 6, 7, 8. How many people did Jeri shake hands with?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b0bde52bff805a4843b12baef43ac146e303c43a6ffdf2660f3d8fc7f084295a | At a party with Mr and Mrs Smith and five other married couples, each person shakes hands with others but not with their spouse. At the end of the hour, Mr Smith asks everyone else how many handshakes they made, and each person gives a different number. How many handshakes did Mrs Smith make? | [
"5"
] | [
"At a party with Mr and Mrs Smith and five other married couples, each person shakes hands with others but not with their spouse. At the end of the hour, Mr Smith asks everyone else how many handshakes they made, and each person gives a different number. How many handshakes did Mrs Smith make?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-032ee4c648bd28a5f19c861a2ccdd0db42a5d71e10308dd5bbe61474715ba5f7 | At a party with Mr. and Mrs. Lewis and three other couples, several handshakes took place. No one shook hands with themselves or their spouse, and no two people shook hands more than once. Each person told Mr. Lewis the number of handshakes they made, and all answers were different. How many handshakes did Mr. and Mrs.... | [
"3"
] | [
"At a party with Mr. and Mrs. Lewis and three other couples, several handshakes took place. No one shook hands with themselves or their spouse, and no two people shook hands more than once. Each person told Mr. Lewis the number of handshakes they made, and all answers were different. How many handshakes did Mr. and... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-397fef754b37a48db94ce67cff08d9d4af4c78bed4e3acc5a3721ca8e760b38c | At a party with Professor McBrain, his wife April, and four other couples, each person shakes hands with a different number of people, excluding their partner. How many people shook hands with April? | [
"4"
] | [
"At a party with Professor McBrain, his wife April, and four other couples, each person shakes hands with a different number of people, excluding their partner. How many people shook hands with April?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-84fa0bf725ef6dc2c36f5f6192b69dfedbb6cdfb383185fea7f775e1db6e31b0 | At a party with \( n \) individuals (where \( n > 1 \)), each person has a different book. Any two people can exchange their books. After several exchanges, it is known that any two people have exchanged their books at least once, and each person ends up with their original book. Find the minimum possible number of exc... | [
"\\( n \\) if \\( n \\) is even, \\( n + 1 \\) if \\( n \\) is odd."
] | [
"At a party with \\( n \\) individuals (where \\( n > 1 \\)), each person has a different book. Any two people can exchange their books. After several exchanges, it is known that any two people have exchanged their books at least once, and each person ends up with their original book. Find the minimum possible numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-03e0debecfaa2864f20dc68c428969acffc0e83b9497998bedc7f4b7b7a2a97b | At a party with \( n \) married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how man... | [
"n-1"
] | [
"At a party with \\( n \\) married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd23ac29f45a1af7935f27365cf59d45d2af889fa20243bbafc3ab24bcf76131 | At a party with \( n \) married couples present, various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening, Mr. Jones asked everyone else, including his wife, how many hands he or she... | [
"n-1"
] | [
"At a party with \\( n \\) married couples present, various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening, Mr. Jones asked everyone else, including his wife, how many hands he... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12fd826551ffed6b81faf0ff84a780f4feed1818fb6635ca4b26b60e812dee69 | At a party with \( n \) people, each person has at least one friend among the others, and any two persons that have the same number of friends have no common friends. Find the least odd value of \( n \) for which this is possible. | [
"5"
] | [
"At a party with \\( n \\) people, each person has at least one friend among the others, and any two persons that have the same number of friends have no common friends. Find the least odd value of \\( n \\) for which this is possible."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-b50c0585fbf024384c82c21bce1b6bd49fe5ea3a772cdf6420152d01a221f759 | At a party with \( n \) people, it is known that among any three people, there are at least two who know each other. Determine the smallest value of \( n \) such that there must be a group of four people where every pair within the group knows each other. Find \( n \). | [
"9"
] | [
"At a party with \\( n \\) people, it is known that among any three people, there are at least two who know each other. Determine the smallest value of \\( n \\) such that there must be a group of four people where every pair within the group knows each other. Find \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf68148736ba8caf71072a34710aef9fabc679ee7f260a8f7342c3d56463269d | At a party with \( n \ge 3 \) guests, some guests know each other. There is no quartet of guests \( a, b, c, d \) such that \( a \) knows \( b \), \( b \) knows \( c \), \( c \) knows \( d \), and \( d \) knows \( a \), but \( a \) does not know \( c \) and \( b \) does not know \( d \). A nonempty set of guests \( X \... | [
"9"
] | [
"At a party with \\( n \\ge 3 \\) guests, some guests know each other. There is no quartet of guests \\( a, b, c, d \\) such that \\( a \\) knows \\( b \\), \\( b \\) knows \\( c \\), \\( c \\) knows \\( d \\), and \\( d \\) knows \\( a \\), but \\( a \\) does not know \\( c \\) and \\( b \\) does not know \\( d \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3677ebb13cbaf6e8da7c6fb220bff578b78bb8d6562b228037063cef98f14e9 | At a party with \( n \geq 4 \) people, if every 3 people have exactly 1 common friend, how many different values can \( n \) take? | [
"\\(\\text{Infinitely many}\\)"
] | [
"At a party with \\( n \\geq 4 \\) people, if every 3 people have exactly 1 common friend, how many different values can \\( n \\) take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b7de0ae61cce4e90fca92dcc7a1ec1ca0013e65090c9a6255ddfb01ca4027f7d | At a party with \(n\) people, if two people have the same number of friends, then they do not have any friends in common. Determine the minimum number of people at the party who know exactly one other person, assuming \(n \geq 3\). | [
"2"
] | [
"At a party with \\(n\\) people, if two people have the same number of friends, then they do not have any friends in common. Determine the minimum number of people at the party who know exactly one other person, assuming \\(n \\geq 3\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-476976b9ea3419c9949a461717286ac9bcf1221113c035254423127be71f405d | At a party with an infinite number of people where each pair of people either knows or does not know each other, define \( f(n) \) as the minimum number of people required to guarantee the existence of a subset of \( n \) people such that either all of them know each other or none of them know each other. Find \( f(5) ... | [
"43"
] | [
"At a party with an infinite number of people where each pair of people either knows or does not know each other, define \\( f(n) \\) as the minimum number of people required to guarantee the existence of a subset of \\( n \\) people such that either all of them know each other or none of them know each other. Find... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-35769ac49f17938da5113f1d1b2dade0ced05dc112f272f7a02e79d9f34dc117 | At a party with eight guests, each participant can't talk with at most three other persons. Suppose the guests are numbered from 1 to 8. Let $S$ be the sum of the number of guests each person can talk to. Calculate the value of $S$. | [
"32"
] | [
"At a party with eight guests, each participant can't talk with at most three other persons. Suppose the guests are numbered from 1 to 8. Let $S$ be the sum of the number of guests each person can talk to. Calculate the value of $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-befaa56d44dcd1bad8016b572564042b382741ba275c6e7e735b9a6fb1d654f3 | At a party with four other couples (including Mr and Mrs Jones), some people shook hands, but no partners shook hands with each other, no two people shook hands more than once, and no one shook hands with themselves. Mr Jones asked everyone else how many times they had shaken hands and received a different answer from ... | [
"4"
] | [
"At a party with four other couples (including Mr and Mrs Jones), some people shook hands, but no partners shook hands with each other, no two people shook hands more than once, and no one shook hands with themselves. Mr Jones asked everyone else how many times they had shaken hands and received a different answer ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
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